A158734 a(n) = 70*n^2 + 1.

1, 71, 281, 631, 1121, 1751, 2521, 3431, 4481, 5671, 7001, 8471, 10081, 11831, 13721, 15751, 17921, 20231, 22681, 25271, 28001, 30871, 33881, 37031, 40321, 43751, 47321, 51031, 54881, 58871, 63001, 67271, 71681, 76231, 80921, 85751, 90721, 95831, 101081, 106471

Offset

0,2

Comments

The identity (70*n^2 + 1)^2 - (1225*n^2 + 35)*(2*n)^2 = 1 can be written as a(n)^2 - A158733(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: -(1 + 68*x + 71*x^2)/(x-1)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

From Amiram Eldar, Mar 22 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(70))*Pi/sqrt(70) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(70))*Pi/sqrt(70) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 71, 281}, 50] (* Vincenzo Librandi, Feb 20 2012 *)

Prog

(Magma) I:=[1, 71, 281]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012
(PARI) for(n=0, 40, print1(70*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 20 2012

Crossrefs

Cf. A005843, A158733.

Sequence in context: A349084 A140856 A114992 * A126021 A142548 A158738

Adjacent sequences: A158731 A158732 A158733 * A158735 A158736 A158737

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 25 2009

Extensions

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved